solve the equation on the interval (0leq\theta<2pi).\n(\tanleft(\frac{\theta}{2}-\frac{pi}{3}\right)=1)\nwhat…

solve the equation on the interval (0leq\theta<2pi).\n(\tanleft(\frac{\theta}{2}-\frac{pi}{3}\right)=1)\nwhat are the solutions in the interval (0leq\theta<2pi)? select the correct choice and fill in any answer boxes in your choice below\noa. the solution set is\n(simplify your answer. type an exact answer, using (pi) as needed. type your answer in radians. use integers or fractions for answers as needed.)\nob. there is no solution.

solve the equation on the interval (0leq\theta<2pi).\n(\tanleft(\frac{\theta}{2}-\frac{pi}{3}\right)=1)\nwhat are the solutions in the interval (0leq\theta<2pi)? select the correct choice and fill in any answer boxes in your choice below\noa. the solution set is\n(simplify your answer. type an exact answer, using (pi) as needed. type your answer in radians. use integers or fractions for answers as needed.)\nob. there is no solution.

Answer

Explanation:

Step1: Solve for (\frac{\theta}{2}-\frac{\pi}{3})

We know that if (\tan x = 1), then (x=\frac{\pi}{4}+k\pi), (k\in\mathbb{Z}). So, (\frac{\theta}{2}-\frac{\pi}{3}=\frac{\pi}{4}+k\pi).

Step2: Solve for (\theta)

First, add (\frac{\pi}{3}) to both sides: (\frac{\theta}{2}=\frac{\pi}{4}+\frac{\pi}{3}+k\pi). Find a common - denominator for the right - hand side: (\frac{\pi}{4}+\frac{\pi}{3}=\frac{3\pi + 4\pi}{12}=\frac{7\pi}{12}). So, (\frac{\theta}{2}=\frac{7\pi}{12}+k\pi). Then multiply both sides by 2: (\theta=\frac{7\pi}{6}+2k\pi).

Step3: Find solutions in the interval (0\leq\theta\lt2\pi)

When (k = 0), (\theta=\frac{7\pi}{6}). Since when (k = 1), (\theta=\frac{7\pi}{6}+2\pi=\frac{7\pi + 12\pi}{6}=\frac{19\pi}{6}\gt2\pi).

Answer:

A. The solution set is (\left{\frac{7\pi}{6}\right})